COMPONENTS OF THE INVOLUTION MODULE IN BLOCKS WITH CYCLIC OR KLEIN-FOUR DEFECT

GROUP

JOHN MURRAY

Abstract. Each p-block of a finite group has an associated defect group, which is a p-subgroup of the group. Each real 2-block has, in addition, an associated extended defect group, which is is a 2subgroup of the group that contains a defect group as a subgroup of index ≤ 2. We consider the possible extended defect groups of a real 2-block that has a cyclic or a Klein-four defect group. In each case we describe the modules in the block that are components in the permutation module of the group acting by conjugation on its involutions. We also determine the Frobenius-Schur indicators of the irreducible characters in the block.

1. Introduction

Let G be a finite group and let O be a complete discrete valuation ring that has a fraction field F of characteristic 0 and residue field k := O/J(O) of characteristic p > 0. We assume that F and k are splitting fields for all subgroups of G. These are standard assumptions in [15]. Mostly, but not exclusively, p = 2. We use R to denote either of the rings O or k. Now G acts by conjugation on

Ω := {g ∈ G | g^{2} = 1_{G}}.

We call the resulting permutation module RΩ the involution module of G. In this paper we classify the components (indecomposable dirrect summands) of RΩ that belong to real 2-blocks of G whose defect groups are cyclic or Klein-four groups. This enables us to determine the Frobenius-Schur indicators of the irreducible characters in such blocks.

As usual G is a right G × G-set via x·(g_{1}, g_{2}) := g_{1}^{−1}xg_{2}, for all x ∈ G and g_{1}, g_{2} ∈ G. If H is a subgroup of G, then ΔH := {(h, h) | h ∈ H} is the diagonal of H in G × G, and if h ∈ H, then Cl_{H}(h) is the Hconjugacy class that contains h. We simplify this to Cl(h), if H = G.

Date: September 25, 2006.

1991 Mathematics Subject Classification. 20C20.

Key words and phrases. Frobenius-Schur indicator, p-permutation module, involutions, cyclic defect, Klein-four defect.

The wreath product G ≀ Σ is a split extension of G × G by the cyclic group Σ = ⟨σ⟩ of order 2. Here (g_{1}, g_{2})σ = σ(g_{2}, g_{1}), for all g_{1}, g_{2} ∈ G. We extend G to a right G ≀ Σ-set by defining x · σ := x^{−1}, for all x ∈ G (see [13]). The map g → (g, g^{−1})σ, establishes a G ≀ Σ-set isomorphism between G and the G ≀ Σ-conjugacy class Cl_{G≀Σ}(σ) of σ. So RG is isomorphic to the induced module (R_{ΔG×Σ})↑^{G≀Σ}, as C_{G≀Σ}(σ) = ΔG×Σ.

Throughout this paper B is a p-block of G. This means that B is a G × G-component of RG. According to J. A. Green, there is a psubgroup D of G such that ΔD is a vertex of B, as G×G-module. This D is also a defect group of B, in the sense of R. Brauer. There is a set Irr(B) of irreducible complex G-characters and a set of RG-modules attached to B. If M is a RG-module, we let MB denote the sum of all submodules of M that belong to B. In particular

RΩB is the sum of all submodules of RΩ that belong to B.

The contragradient block B^{o} is the p-block of G that is the image of B under σ. Thus B = B + B^{o} is a G ≀ Σ-component of RG. We let B̂ be the p-block of G ≀ Σ that contains this component. The block B^{o} contains the complex conjugates of the irreducible characters in B and the duals of the RG-modules in B. We say that B is real if B = B^{o}.

For the rest of this section p = 2 and B is a real 2-block of G. So B is a G≀Σ-component of RG. By [14, Lemma 10], there exists e ∈ N_{G}(D), with e^{2} ∈ D, such that ΔD⟨Δe σ⟩ is a vertex of B as G ≀ Σ-module. We call the 2-group E := D⟨e⟩ an extended defect group of B. If B is the principal 2-block of G then E = D; otherwise [E : D] = 2.

We justify the adjective ‘extended’ as follows: There is a conjugacy class of G which occurs in the support of the block idempotent of B and on which the central character of B does not vanish. Any such class is called a defect class of B. It is known that there exists g ∈ G in a defect class of B such that the defect group D is a Sylow 2-subgroup of the centralizer C_{G}(g) of g in G. According to [10], B has a real defect class. By [14] there exists g ∈ G in this class such that E is a Sylow 2-subgroup of the extended centralizer C_{G}^{∗}(g) of g in G. Here C_{G}^{∗}(g) := {x ∈ G | x^{−1}gx = g^{±1}}.

The Frobenius-Schur indicator of a generalized character χ of G is the integer ϵ(χ) := \frac{1}{|G|} ∑_{g∈G} χ(g^{2}). Suppose that χ is irreducible. Then ϵ(χ) ∈ {0, ±1}. Moreover ϵ(χ) = 0, if χ is not real-valued, and ϵ(χ) = +1 if χ is the character of a real representation. Otherwise ϵ(χ) = −1. The structure of E affects the Frobenius-Schur indicators of the irreducible characters in B, as the following two lemmas indicate.

Lemma 1.1. B has an irreducible character with Frobenius-Schur indicator −1 and height zero if and only if E/D′ does not split over D/D′.

Proof. This is a result of R. Gow. See Theorem 5.6 in [9]. □

Lemma 1.2. The following are equivalent:

(i) E splits over D; (ii) RΩB ≠ 0;

(iii) ∑_{χ∈Irr(B)}ϵ(χ)χ(1) ≠ 0.

Proof. This is Theorem 2 in the author’s paper [14]. □

We call B a strongly real 2-block of G, if it satisfies these conditions. We shall prove a number of general results in Section 2, in particular

the following two theorems about real 2-blocks:

Theorem 1.3. Suppose that B is strongly real. Then there is a component of RΩB that has vertex C_{D}(x), for some x ∈ Ω such that E = D⟨x⟩. Moreover, each component of RΩB has a vertex contained in C_{D}(y), for some y ∈ Ω such that E = D⟨y⟩.

In the terminology of Brauer, a root of B (in D C_{G}(D)) is a 2-block β of D C_{G}(D) such that β^{G} = B. Each root of B has defect group D, and the roots of B form a single N_{G}(D)-orbit under conjugation.

Theorem 1.4. B has a real root if and only if E ≤ D C_{G}(D).

The remainder of the paper concentrates on real 2-blocks that have a cyclic or Klein-four defect group. For such blocks we have:

Lemma 1.5. Suppose that D is cyclic or a Klein-four group. Then B has an irreducible character with Frobenius-Schur indicator −1 if and only if E does not split over D.

Proof. This follows from Lemma 1.1 and the fact that all irreducible characters in B have height zero, as detailed below. □

E. C. Dade described the decomposition matrices, and much of the character theory of blocks with cyclic defect group in [6]. In particular, if D is cyclic, then there is a unique irreducible kG-module S in B, and there are |D| ordinary irreducible G-characters in B. It is also known that for each i = 1, 2, …, |D|, there is a unique indecomposable RGmodule S_{i} in G such that S_{i} has i composition factors. The subgroup of index gcd(|D|, i) in D is a vertex of S_{i}.

Here are names for 2-groups with maximal subgroups that are cyclic: • D_{2^{n}} := ⟨a, b | a^{2^{n−1}} = b^{2} = 1, a^{b} = a^{−1}⟩ is a dihedral group; • Q_{2^{n}} := ⟨a, b | a^{2^{n−1}} = 1, b^{2} = a^{2^{n−2}}, a^{b} = a^{−1}⟩ is a generalized

quaternion group;

• SD_{2^{n}} := ⟨a, b | a^{2^{n−1}} = b^{2} = 1, a^{b} = a^{2^{n−2}−1}⟩ is a semidihedral group;

• M_{2^{n}} := ⟨a, b | a^{2^{n−1}} = b^{2} = 1, a^{b} = a^{2^{n−2}+1}⟩ is a modular group. The term ‘modular 2-group’ is used occasionally in finite group theory. A recent example can be found in [8].

Here is our main result about real 2-blocks with a cyclic defect group:

Theorem 1.6. If D is cyclic, then one of the following is true:

(i) E is cyclic. Two irreducible characters in B are real-valued; one of these has Frobenius-Schur indicator −1.

(ii) E ≅ D × ℤ_{2}. Two irreducible characters in B are real-valued

and kΩB ≅ S ⊕ S. (iii) E is a dihedral group. All irreducible characters in B are real-

valued and kΩB ≅ S_{|D|/2} ⊕ S_{|D|/2}. (iv) E is a semi-dihedral group. Half of the irreducible characters

in B are real-valued and kΩB ≅ S_{|D|/2}.

(v) E is a modular group. Two irreducible characters in B are real-valued and kΩB ≅ S_{2}.

(vi) E is a generalized quaternion group. All irreducible characters in B are real-valued; half of these have Frobenius-Schur indicator −1.

We use rad(M) to denote the radical and soc(M) to denote the socle of a kG-module M. The Loewy series of M is M ⊃ rad(M) ⊃ rad^{2}(M) ⊃ . . .. We use P_{M} to denote the projective cover of M. If A, B, … are irreducible modules, then U(A, B, …) is a uniserial module with successive Loewy factors A, B, . . ..

Now suppose that D is a Klein-four group. Then D = ⟨s, t⟩, where s, t and st are involutions. R. Brauer showed in [4, Section 7] that B has four ordinary irreducible characters and either one or three irreducible modules. K. Erdmann [7, Theorem 4] Green correspondence and Auslander-Reiten theory to describe the possible decomposition matrices of B.

We list the irreducible characters in B as χ_{1}, χ_{2}, χ_{3}, χ_{4}. Let β be a root of B and let I = {n ∈ N_{G}(D) | β^{n} = β} be the inertial group of β in N_{G}(D). Then [I : C_{G}(D)] is an odd integer, called the inertial index of B. If the inertial index is 1 then B has a single irreducible module S; if the inertial index is 3 then B has three irreducible modules S, X and Y. In the latter case s, t and st are conjugate in G and C_{G}(s) has a unique 2-block b_{1} such that b_{1}^{G} = B. Moreover, b_{1} has inertial index 1. Let ψ_{1} denote the unique irreducible Brauer character in b_{1}. We obtain the following by combining results from [4] and [7].

Lemma 1.7. If D is a Klein-four group, there are three block types:

(I) B has inertial index 1 and rad(P_{S})/S ≅ S ⊕ S. All decompo-

sition numbers are 1; (II) B has inertial index 3 and

rad(P_{S})/S ≅ X ⊕ Y, rad(P_{X})/X ≅ Y ⊕ S, rad(P_{Y})/Y ≅ S ⊕ X.

Thus D_{B} = [\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}].

There is a sign δ = ±1 such that for all 2-regular elements g ∈ C_{G}(s), we have χ_{1}(gs) = χ_{3}(gs) = χ_{4}(gs) = δψ_{1}(g) and χ_{2}(gs) = −δψ_{1}(g);

(III) B has inertial index 3 and

rad(P_{S})/S ≅ U(X,S, Y) ⊕ U(Y, S, X),
rad(P_{X})/X ≅ U(S, Y, S), rad(P_{Y})/Y ≅ U(S, X, S).

Thus D_{B} = [\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}].

There is a sign δ = ±1 such that for all 2-regular elements g ∈ C_{G}(s), we have χ_{1}(gs) = χ_{2}(gs) = δψ_{1}(g) and χ_{3}(gs) = χ_{4}(gs) = −δψ_{1}(g).

We give our main results for real 2-blocks with Klein-four defect groups in three theorems:

Theorem 1.8. Let B be of type (I). Then one of the following holds:

(i) E ≅ ℤ_{4} × ℤ_{2}. All four irreducible characters in B are realvalued; two of these have Frobenius-Schur indicator −1.

(ii) E ≅ ℤ_{2}^{3}. All four irreducible characters in B are real-valued

and kΩB ≅ S^{4}; (iii) E ≅ D_{8}. Two irreducible characters in B are real-valued and

kΩB ≅ U(S, S) is indecomposable with a vertex of order 2.

Theorem 1.9. Let B be of type (II). Then one of the following holds:

(i) E ≅ ℤ_{2}^{3}. Two irreducible characters in B are real-valued and kΩB ≅ S^{2} ⊕ X ⊕ Y; (ii) E ≅ D_{8}. All four irreducible characters in B are real-valued and kΩB is indecomposable with a vertex of order 2. Moreover hd(kΩB) ≅ rad(kΩB) ≅ S ⊕ X ⊕ Y.

Theorem 1.10. Let B be of type (III). Then one of the following holds:

(i) E ≅ ℤ_{2}^{3}. All four irreducible characters in B are real-valued and kΩB ≅ S^{2} ⊕ U(X, S, Y) ⊕ U(Y, S, X);

(ii) E ≅ D_{8}. Two irreducible characters in B are real-valued and kΩB is indecomposable with a vertex of order 2. Moreover hd(kΩB)≅soc(kΩB)≅S and rad(kΩB)/ soc(kΩB)≅X ⊕ Y.

2. Preliminary results

In this section we describe some results about blocks that do not depend on the isomorphism type of their defect groups. Generally B is a p-block of G and D is a defect group of B. If p = 2, and B is a real 2-block, then E is an extended defect group of B that contains D. Each g ∈ G can be written uniquely as g = g_{p}g_{p′} = g_{p′}g_{p}, for g_{p} a p-element in G and g_{p′} a p′-element in G. In fact, g_{p}, g_{p′} ∈ ⟨g⟩.

Lemma 2.1. Suppose that D ⊴ G and that B has a real root and inertial index 1. Then the number of real irreducible characters in B equals the number of real irreducible characters of D.

Proof. Let β be a real root of B. By [15, 5.8.14], there is a unique irreducible Brauer character ψ in β, and the irreducible characters in β are {X_{λ} | λ ∈ Irr(D)}. Here if λ ∈ Irr(D),

X_{λ}(g) = \{ \begin{matrix}ψ(g_{p′})λ(g_{p}), & if g_{p} ∈ D, \\ 0, & otherwise, \end{matrix} for all g ∈ D C_{G}(D).

Since ψ is real, the character X_{λ} is real if and only if λ is real.

Let g ∈ G\ DC_{G}(D). Then by hypothesis β^{g} ≠ β. So X_{λ}^{g} ≠ X_{λ}. It follows that X_{λ}↑^{G} is an irreducible character in B. Moreover, the set of irreducible characters in B is {X_{λ}↑^{G} | λ ∈ Irr(D)}. Clearly X_{λ}↑^{G} is real if X_{λ} is real. Suppose that X_{λ} is not real. Then X_{λ} and its complex conjugate \overline{X_{λ}} both belong to β. It follows that they are not G-conjugate. So X_{λ}↑^{G} is not real. The Lemma follows from this. □

We state without proof the next two results.

Lemma 2.2. Let N ⊴ G, let M be a kN-module and let L be a kG/Nmodule. Then M↑^{G} ⊗ L ≅ dim(L) M↑^{G}.

Lemma 2.3. Let p = 2 and let I ≤ H ≤ G with [H : I] = 2. Then there is a short exact sequence of RG-modules

0 → R_{H}↑^{G} → R_{I}↑^{G} → R_{H}↑^{G} → 0.

Thus the composition multiplicity of an irreducible kG-module in k_{I}↑^{G} is twice its multiplicity in k_{H}↑^{G}.

A p-permutation RG-module is an RG-module whose restriction to a p-subgroup of G is a permutation module. An indecomposable RGmodule is a p-permutation module if and only if it has a source module that is trivial. By [15, 4.8.9] if M is a p-permutation kG-module then there is a unique p-permutation OG-module M̃ such that M =M̃ ⊗_{O} k. We refer to the character of M̃ as the O-character of M.

Lemma 2.4. Let X be a G-set and let V be a p-subgroup of G. Green correspondence establishes a multiplicity preserving correspondence:

{G-components of RX that have vertex V } ↕

{N_{G}(V)/V -components of R C_{X} (V) that are projective}.

If M ↔ N under this correspondence, and M belongs to B, then N belongs to a p-block b of N_{G}(V) such that b^{G} = B.

Proof. This is Theorem 3.2 of [5]. □

Lemma 2.5. Let M be an indecomposable p-permutation RG-module let χ_{M} be the O-character of M, and let g be a p-element in G. Then

χ_{M}(g) \{ \begin{matrix}∈ ℤ_{+}, & if g belongs to some vertex of M; \\ = 0, & otherwise. \end{matrix}

Proof. The character value χ_{M}(g) is the number of trivial components of M↓_{⟨g⟩}. See [12] for details. □

The following is a variant of Lemma 9.7 of [1].

Lemma 2.6. Let M be an indecomposable p-permutation RG-module and let H be a subgroup of G. Let V be a vertex of M such that V ∩H is not a proper subgroup of U ∩H, for any vertex U of M. Then some component of M↓_{H} has vertex V ∩ H.

Proof. By hypothesis R_{V} is a component of M↓_{V}. So R_{V ∩H} is a component of M↓_{V ∩H}. Choose a component S of M↓_{H} such that R_{V ∩H} is a component of S↓_{V ∩H}. Now V ∩H is a vertex of R_{V ∩H}. So V ∩H ≤ W, for some vertex W of S, by Lemma 4.3.4 of [15]. Similarly W ≤ U, for some vertex U of M. Thus V ∩ H ≤ W ≤ U ∩ H. The assumption on V ∩ H forces V ∩ H = U ∩ H. We conclude that V ∩ H = W. □

The Frobenius automorphism of k is given by Fr(λ) = λ^{p}, for all λ ∈ k. If M is a kG-module, then we can apply Fr to the entries in the matrices representing the elements of G in GL(M), to get a kG-module M^{Fr} called the Frobenius twist of M. If M has Brauer character ϕ, then M^{Fr} has Brauer character ϕ^{Fr} where ϕ^{Fr}(g) = ϕ(g^{p}), for all p-regular g ∈ G. Clearly M is indecomposable/irreducible if and only if M^{Fr} is indecomposable/irreducible. Moreover, as Fr commutes with induction, if M is indecomposable then M and M^{Fr} share the same vertices. Let B^{Fr} be the p-block of G that contains the Frobenius twists of all the kG-modules in B. Then B and B^{Fr} have the same defect classes and defect groups. Moreover Irr(B^{Fr}) = {χ^{Fr} | χ ∈ Irr(B)}, where χ^{Fr}(g) := χ(g_{p}g_{p′}^{p}), for all g ∈ G.

Lemma 2.7. There is a multiplicity and vertex preserving correspondence between the components of RΩB and the components of RΩB^{Fr}.

Proof. This follows from the discussion above and the fact that kΩ = (kΩ)^{Fr}. □

Lemma 2.8. Suppose that B is a real 2-block of G. Then

B↓_{ΔG×Σ} = RΩB^{Fr} ⊕ X,

where no component of X has a vertex that contains Σ.

Proof. Here we are identifying the 2-blocks of G and ΔG × Σ. Now kG↓_{ΔG×Σ} = kΩ ⊕ k(G\Ω), where kΩ is centralized by Σ, and no component of k(G\Ω) has a vertex that contains Σ. Suppose that M is a component of the inflation of kΩB^{Fr} to ΔG × Σ. Then M has a vertex that contains Σ. But C_{G≀Σ}(Σ) = ΔG × Σ, and (B^{Fr})^{G≀Σ} = B̂, by Lemma 16 of [14]. It then follows from Lemma 3.7a of [11] that M is a direct summand of kGB̂ = B. □

Proof of Theorem 1.3. Choose e ∈ E so that E = D⟨e⟩ and set H := ΔG×Σ. Each vertex of B is ΔG-conjugate to (ΔD⟨Δeσ⟩)^{(1,y)}, for some y ∈ G. Suppose that such a vertex contains σ. Then there exists d_{0} ∈ D such that σ = (Δ(d_{0}e)σ)^{(1,y)}. But (Δ(d_{0}e)σ)^{(1,y)} = (d_{0}ey, y^{−1}d_{0}e)σ. So y = d_{0}e, whence E = D⟨y⟩, and y ∈ Ω. Moreover

(1)  (ΔD⟨Δeσ⟩)^{(1,y)} ∩ H = Δ C_{D}(y) × Σ.

Let x ∈ Ω be such that E = D⟨x⟩ and C_{D}(x) ≮ C_{D}(y), for all y ∈ Ω such that E = D⟨y⟩. Set V := (ΔD⟨Δeσ⟩)^{(1,x)}. Then V is a vertex of B, and (1) implies that V ∩ H ≮ U ∩ H, for any vertex U of B. Lemma 2.6 shows that there is a component of B↓_{ΔG×Σ} that has vertex V ∩ H = Δ C_{D}(x) × Σ. We conclude from Lemma 2.8 that RΩB^{Fr} has a component that has vertex C_{D}(x). The first statement now follows from Lemma 2.7.

Let M be a component of RΩB, as RG-module, and let W ≤ D be a vertex of M. Then M has vertex ΔW × Σ, as RΔG × Σ-module.

Lemma 2.8 implies that M^{Fr} is a component of B↓_{ΔG×Σ}. Using (1), there exists y ∈ E such that E = D⟨y⟩ and W ≤ C_{D}(y). This proves the second statement. □

The vertices of components of RΩ also have a lower bound:

Lemma 2.9. Suppose that p = 2. Then each projective component of RΩ belongs to a real 2-block with a trivial defect group.

Proof. This is the main result of [13]. □

We will make use of the following special result in Sections 4 and 5:

Lemma 2.10. Suppose that B is a strongly real 2-block such that D⊴G and E ≤ D C_{G}(D). Set Z := Ω(Z(D)). Consider kZ as Z ⋊ G/Dmodule, with Z acting regularly and G/D acting by conjugation. Then there is a self-dual irreducible kG-module T in B such that the module

∑_{i≥0} T ⊗ rad^{i}(kZ)/rad^{i+1}(kZ)

is a direct summand of kΩB, and all its components have vertex D.

Proof. We identify the 2-blocks of G, ΔG and ΔG × Σ. The group G acts by conjugation on Z, and D ≤ C_{G}(Z). So there is a well-defined action of G/D on Z. The group Z acts regularly on kZ. In this way kZ is a module for the semi-direct product group Z ⋊ G/D.

Choose e ∈ Ω_{C_{G}(D)} such that E = D × ⟨e⟩. Then (Δeσ)^{(1,e)} = σ, and ΔD^{(1,e)} = ΔD. So V := ΔD × Σ is a vertex of B. Now

ΔG×Σ ≤ N_{G≀Σ}(V) ≤ N_{G≀Σ}(ΔD) = ΔG(C_{G}(D) ≀ Σ).

Let c ∈ C_{G}(D) be such that (1, c) normalizes V. Then σ^{(1,c)} = (c, c^{−1})σ belongs to V. This forces c ∈ Z. So N_{G≀Σ}(V) = ΔG(Z ≀ Σ).

By Lemma 2.4, there is a component fB of B↓_{N_{G≀Σ}(V)} that has vertex V and kernel containing V. In particular all components of fB↓_{ΔG×Σ} have vertex V. So by Lemma 2.8, the module fB↓_{ΔG} is a direct summand of kΩB^{Fr}, and all of its components have vertex D.

Now N_{G≀Σ}(V)/V ≅ Z ⋊ G/D. So we can regard fB as a projective indecomposable Z ⋊ G/D-module. Let S be the irreducible socle of fB. Then S is self-dual, as B and hence fB are self-dual. Proposition 18.4 of [1] implies that

fB↓_{ΔG/ΔD} ≅ ∑_{i≥0} S ⊗_{k} rad^{i}(kZ)/rad^{i+1}(kZ).

The result now follows from Lemma 2.7, and the fact that kZ = (kZ)^{Fr}. □

The following result is similar to Lemma 13.7 in [1].

Lemma 2.11. Let P be a p-subgroup of G. Then

B↓_{P C_{G}(P)≀Σ} = ∑(b + b^{o}) ⊕ X,

where b ranges over the p-blocks of P C_{G}(P) such that b^{G} = B and no component of X has a vertex that contains ΔP.

Proof. Set H := P C_{G}(P). Then kG↓_{H≀Σ} = kH ⊕ k(G\H), as kH ≀ Σmodules. Now ΔP does not centralize any element of G\H. So by Lemma 2.4, no component of k(G\H) has a vertex that contains ΔP.

The kH ≀ Σ-components of kH have the form b + b^{o}, where b is a p-block of H. Let b be such a block and let b̂ be the p-block of H ≀ Σ that contains b + b^{o}. As P is a normal subgroup of H, it is contained in a defect group of b, by [1, 13.6]. So ΔP is contained in a vertex of b + b^{o}, as kH ≀ Σ-module. Now C_{G≀Σ}(ΔP) = C_{G}(P) ≀ Σ ≤ H ≀ Σ. So b̂^{G≀Σ} = B̂, by Lemma 3.7a of [11]. It follows from this that b^{G} = B. □

The O-character χ_{B} of B is computed in Lemma 8 of [14] as:

(2) \begin{aligned}χ_{B} ((g_{1}, g_{2})) & = ∑_{χ∈Irr(B)} χ(g_{1}^{−1})χ(g_{2}), \\ χ_{B} ((g_{1}, g_{2})σ) & = ∑_{χ∈Irr(B)} ϵ(χ)χ(g_{1}g_{2}), \end{aligned} for all g_{1}, g_{2} ∈ G.

Our next result appears as Theorem (4B) in R. Brauer’s paper [3]:

Corollary 2.12. Let g ∈ G. Then

(3)  ∑_{χ∈Irr(B)} ϵ(χ)χ(g) = ∑_{b} ∑_{χ̂∈Irr(b)} ϵ(χ̂)χ̂(g),

where b ranges over the p-blocks of C_{G}(g_{p}) such that b^{G} = B.

Proof. We claim that the p-part of (g, 1)σ ∈ G≀Σ generates a group that contains Δg_{p}. Suppose first that p is odd. There is a unique s ∈ ⟨g_{p}⟩ such that s^{2} = g_{p}. Then Δs is a p-element and ⟨Δg_{p}⟩ = ⟨Δs⟩. Also (Δs)^{−1} (g, 1)σ is a p′-element, because its square is (g_{p′}, g_{p′})σ. So Δs is the p-part of (g, 1)σ. This proves the claim when p is odd. Conversely, suppose that p = 2. Then (g, 1)σ has 2-part (g_{2}, 1)σ. So the claim follows from the fact that Δg_{2} = ((g_{2}, 1)σ)^{2}.

With P = ⟨g_{p}⟩, Lemma 2.11 gives

B↓_{C_{G}(g_{2})≀Σ} = ∑_{b^{G}=B}(b + b^{o}) ⊕ X,

where no component of X has a vertex that contains Δg_{p}. It then follows from the previous paragraph, and Theorem 4.7.4 of [15], that χ_{B} ((tu, 1)σ)) = ∑_{b^{G}=B} χ_{b} ((g, 1)σ)). Equation (3) is a consequence of this and (2). □

The following is a variant of [14, Corollary 15]:

Lemma 2.13. Suppose that B is a real 2-block of G. Let t be a 2element in G. Then ∑_{χ∈Irr(B)} ϵ(χ)χ(t) is a nonnegative integer, which is positive if and only if t is G-conjugate to e^{2}, for some e ∈ E such that E = D⟨e⟩.

Proof. The element (t, 1)σ ∈ G ≀ Σ has 2-power order, as its square is the 2-element Δt ∈ G × G. By (2), χ_{B} ((t, 1)σ) = ∑_{χ∈Irr(B)} ϵ(χ)χ(t). So Lemma 2.5 implies that ∑_{χ∈Irr(B)} ϵ(χ)χ(t) is a non-negative integer.

Suppose that (t, 1)σ belongs to a vertex of B. Then there exist g_{1}, g_{2} ∈ G and e ∈ E such that E = D⟨e⟩ and ((t, 1)σ)^{(g_{1},g_{2})} = (e, e)σ. Thus g_{1}^{−1}tg_{2} = g_{2}^{−1}g_{1} = e, whence e^{2} = t^{g_{1}} is G-conjugate to t.

Conversely, suppose that e^{2} = t^{g_{1}}, where g_{1} ∈ G and E = D⟨e⟩. Then ((t, 1)σ)^{(g_{1},g_{1}e^{−1})} = (e, e)σ. So (t, 1)σ belongs to a vertex of B. The result now follows from Lemma 2.5. □

Lemma 2.14. Let p = 2, let P be a principal indecomposable kGmodule and let Φ be the O-character of P. Then the multiplicity of the irreducible module P/rad(P) as a composition factor of kΩ is ϵ(Φ).

Proof. See [14, Lemma 3] or [16, Lemma 1]. □

A subgroup pair of G is a pair (H, I) such that H ≤ I ≤ G and [I : H] = 2. We say that a pair (J, K) is a subpair of (H, I) if J ≤ H and KH = I. More generally, (J, K) is G-conjugate to a subpair of (H, I) if there exists g ∈ G such that (J^{g}, K^{g}) is a subpair of (H, I). Clearly G-conjugacy is a transitive relation on subgroup pairs.

If B is a real non-principal 2-block of G then (D, E) is a subgroup pair of G. The G-conjugates of (D, E) are called the defect pairs of B.

Lemma 2.15. Suppose that B is a real 2-block of G. Let H ≤ G and let b be an admissible real non-principal 2-block of H such that B = b^{G}. Then each defect pair of b is a subpair of a defect pair of B. In particular, if b is strongly real then B is strongly real, while if D is a defect group of b, then each defect pair of b is a defect pair of B.

Proof. If X ⊆ G, set X^{+} := ∑_{x∈X} x, in kG. Let ω_{B}, ω_{b} be the central character of B, respectively b. Let C be a real defect class of B. Then ω_{b}((C ∩ H)^{+}) = ω_{B}(C^{+}) ≠ 0_{k}. We pair the contribution of each Hconjugacy class in C ∩ H to ω_{b}((C ∩ H)^{+}) with that of its inverse class in C ∩ H. In this way we see that there is a real H-conjugacy class C_{1} ⊆ C ∩ H such that ω_{b}(C_{1}^{+}) ≠ 0_{k}. Let g ∈ C_{1}, let D_{H} be a Sylow 2-subgroup of C_{H}(g) and let E_{H} be a Sylow 2-subgroup of C_{H}^{∗}(g) that contains D_{H}. Theorem 2.1 in [10] implies that some defect pair (D_{b}, E_{b}) of b is a subpair of (D_{H}, E_{H}). But g ∈ C. So (D_{H}, E_{H}) is G-conjugate to a subpair of (D, E). It follows that (D_{b}, E_{b}) is G-conjugate to a subpair of (D, E).

Suppose that b is strongly real. Then C_{1} is a strongly real class of H. So C ⊇ C_{1} is a strongly real class of G. Thus B is strongly real.

Finally, suppose that D is a defect group of b. Then D_{b} is a Sylow 2-subgroup of C_{G}(g) and E_{b} is a Sylow 2-subgroup of C_{G}^{∗}(g). It follows that (D_{b}, E_{b}) is a defect pair of B. □

Proof of Theorem 1.4. Suppose that B has a real root β. Let (D, E_{1}) be a defect pair of β. Lemma 2.15 implies that (D, E_{1}) is a defect pair of B. So there exists an element g ∈ G in a defect class of B such that D is a Sylow 2-subgroup of C_{G}(g) and E_{1} is a Sylow 2-subgroup of C_{G}^{∗}(g). But then g is a real element in C_{G}(D).

Conversely, suppose that E ≤ D C_{G}(D). Lemma 2.11 gives

B↓_{D C_{G}(D)≀Σ} = ∑(β_{1} + β_{1}^{o}) ⊕ X,

where β_{1} ranges over the roots of B, and no component of X has a vertex that contains ΔD. Suppose that E = D⟨e⟩. Then ΔD⟨Δeσ⟩ is a vertex of B, as G ≀ Σ-module. The hypothesis is that ΔD⟨Δeσ⟩ ≤ D C_{G}(D) ≀ Σ. It then follows from Lemma 2.6 that there is a root β_{1} of B such that β_{1} +β_{1}^{o} has vertex ΔD⟨Δeσ⟩. But then β_{1} is a real 2-block of D C_{G}(D). We deduce that so too is its N_{G}(D)-conjugate β. □

3. Cyclic Defect group

In this section B is a real 2-block of G that has a cyclic defect group. We follow Dade’s original approach from [6]. Recall the notation introduced in Section 1. Let d be a generator for the defect group D and suppose that |D| = 2^{n}. Then t := d^{2^{n−1}} is the unique involution in D. Let χ be the Brauer character of the irreducible B-module S. So χ is real-valued. We note that P := S_{2^{n}} is the unique projective indecomposable B-module and S_{2^{n−1}} is the unique indecomposable Bmodule that has ⟨t⟩ as a vertex.

Set H := C_{G}(t). As H ≥ N_{G}(D), Brauer’s first main theorem implies that there is a unique block b of H such that b^{G} = B, and D is a defect group of b. As b is unique it is also real, and Lemma 2.15 implies that each of its defect pairs is a defect pair of B. Let T be the unique irreducible kH-module in b. We may label the indecomposable kH-modules in b as T := T_{1}, T_{2}, …, T_{2^{n}}, where T_{i} is uniserial with i composition factors, each isomorphic to T.

Lemma 3.1. Let V be a nontrivial subgroup of D and let f be the Green correspondence with respect to (G, V, H). Then f(S_{i}) = T_{i} or f(S_{i}) = T_{2^{n}−i}, for each i = 1, 2, …, 2^{n}.

Proof. As usual \overline{End}_{kG}(M) is the quotient of End_{kG}(M) by the subspace of projective endomorphisms, for any module M. We make use of [1, 21.3, 21.4], and the fact that B and b have inertial index 1.

Let 1 ≤ i ≤ 2^{n−1} and let θ be a non-zero G-endomorphism of S_{i}. As i + i − 2^{n} ≤ 0, and as S_{i} has composition length i, [1, 21.3] implies that θ is not a projective homomorphism. Thus End_{kG}(S_{i}) ≅ \overline{End}_{kG}(S_{i}). We deduce from this that \overline{End}_{kG}(S_{i}) has dimension i as k-vector space. The Heller translate of S_{i} is S_{2^{n}−i}. So \overline{End}_{kG}(S_{2^{n}−i}) ≅ \overline{End}_{kG}(S_{i}), by [1, 20.6]. Thus \overline{End}_{kG}(S_{2^{n}−i}) also has dimension i as k-vector space. The same results hold for \overline{End}_{kG}(T_{i}) and \overline{End}_{kG}(T_{2^{n}−i}).

Now let 1 ≤ i ≤ 2^{n}. Then \overline{End}_{kG}(S_{i}) ≅ \overline{End}_{kG}(fS_{i}) by [1, Corollary 4]. The result follows from this and the previous paragraph. □

We can label the irreducible characters in B as X_{λ}, where λ ranges over Irr(D). The characters X_{λ} with λ ≠ 1 are called the exceptional characters in B. The trivial character X_{1} is the unique non-exceptional character in B. If d_{λ,χ} is the multiplicity of χ in X_{λ}, then d_{λ,χ} = 1, for all λ. We use a dash, as in χ′, X′_{λ}, for other objects associated to b.

Lemma 3.2. If λ ∈ Irr(D), then X_{λ} is real if and only if X′_{λ} is real.

Proof. The automorphism group of D is a 2-group. But B has odd inertial index. So C_{G}(D) is the inertial subgroup of a root of B in N_{G}(D).

If g ∈ G is 2-regular then X_{λ}(g) = χ(g) and X′_{λ}(g) = χ′(g) are both real-valued. If g ∈ G is such that g_{2} is G-conjugate to an element of D\{1} then Corollary 1.9 of [6] implies that X_{λ}(g) = ±X′_{λ}(g) (the symbols C_{i}, N_{i} and ϕ_{i} in the statement of that Corollary are the same for B and b, while the symbols ε_{i} and γ_{i} are signs). If g ∈ G is such that g_{2} is not G-conjugate to an element of D then by a theorem of J. A. Green, X_{λ}(g) = 0 and X′_{λ}(g) = 0. The lemma follows easily from these facts. □

Let X be a component of kΩB and let V be a vertex of X that is contained in D. As D ≠ ⟨1⟩, Lemma 2.9 implies that ⟨t⟩ ≤ V. Thus N_{G}(V) ≤ H. Let f be the Green correspondance with respect to (G, V, H). Then by Lemma 2.4, fX is a component of kΩ(H) that belongs to b and that has kernel containing ⟨t⟩.

By the results of section 5.8.3 of [15], there is a unique 2-block \overline{b} of \overline{H} := H/⟨t⟩ that is dominated by b. Now \overline{b} is real. Let g ∈ H belong to a real defect class of b. Then g is 2-regular. The proof of Lemma 5.8.9 of [15] can be adapted to show that C∗ ; \overline{H}(\overline{g}) = \overline{C_{H}^{∗}(g)}. So we may assume that (\overline{D}, \overline{E}) is a defect pair of \overline{b}.

Lemma 3.3. Suppose that B is not the principal block. Let s ∈ Ω be such that kCl(s)B ≠ 0. Then some G-conjugate of s lies in E\D.

Proof. The proof is by induction on |E|. The base case is D = ⟨1⟩ and E is cyclic of order 2. In that case the conclusion follows from Theorem 19 of [14].

By Lemma 2.4 we may assume that s ∈ H, and that k_{C_{H}(s)}↑^{H} has a componext X that belongs to b. Then X can be regarded as a component of k\overline{C_{H}(s)}↑\overline{H} that belongs to \overline{b}. Now \overline{C_{H}(s)} is a subgroup of index ≤ 2 in C\overline{H}(\overline{s}). Thus k C\overline{H}(\overline{s})↑\overline{H} has a composition factor of \overline{b}, using Lemma 2.3. Now k C\overline{H}(\overline{s})↑\overline{H} is a direct summand of kΩ(\overline{H}), as \overline{H}-modules. Moreover, \overline{b} is a real 2-block with cyclic defect group \overline{D} of smaller order than D. By our inductive assumption, \overline{s} is \overline{H}-conjugate to an element of \overline{E}\\overline{D}. It follows from this that s is H-conjugate to an element of E\D. □

Abelian extended defect group

We prove parts (i) and (ii) of Theorem 1.6. So assume that E ≤ C_{G}(D). Theorem 1.4 implies that b has a real root β. As D has two real-valued linear characters, Lemma 2.1 implies that b has two realvalued irreducible characters. We deduce from Lemma 3.2 that B has two real-valued irreducible characters. As all decomposition numbers in B are one, both of these characters have the same degree.

Suppose that E is a cyclic group. Lemma 1.2 implies that kΩB = 0 and ∑_{λ∈Λ} ϵ(X_{λ})X_{λ}(1) = 0. So one of the real-valued irreducible characters in B has Frobenius-Schur indicator +1 and the other has indicator −1.

Now suppose that E ≅ D × ℤ_{2}. Lemma 2.14 implies that kΩB has two composition factors that are isomorphic to S. But Theorem 1.3 implies that kΩB has at least one component with vertex D. The only possibility is that kΩB = S ⊕ S. The O-character of kΩB is the sum of the two real irreducible characters in B.

Note that kΩ(H)b = T ⊕ T. This can be used to show that T is the Green correspondent of S with respect to (G, D, H).

To prove the other parts of Theorem 1.6, we need some information on the elements s ∈ Ω such that kCl(s)B ≠ 0. In case B is the principal 2-block of G, we have E = D, T = k_{H}, and kCl(1_{H})B = k_{H} and kCl(t)B = k_{H}.

Suppose that B is not the principal 2-block of G. Then E = D ×⟨e⟩, where e is an involution. Moreover, e and te are the only involutions in E\D. We claim that S occurs twice as a component of kCl(e), if e and te are conjugate in G. Otherwise it occurs once as a component of each of kCl(e) and kCl(te). For, b covers a unique real 2-block of H/D that has defect pair (D/D, E/D). By Theorem 19 of [14], S occurs once as a component of k_{C_{H/D}(eD)}↑^{H/D}. Now e and te are the only involutions in E\D. So N_{H}⟨t, e⟩ is the inverse image of C_{H/D}(eD) in H. We deduce that S occurs once as a component of k_{N_{H}⟨t,e⟩}↑^{H}.

Say that e is H-conjugate to te. Then [N_{H}⟨t, e⟩ : C_{H}(e)] = 2. Lemma 2.3 implies that T occurs twice as a composition factor of kCl_{H}(e). But then kCl_{H}(e)b = T ⊕T, by our work above. It then follows from Lemma 2.4 that kCl(e)B = S ⊕ S,

Say that e is not H-conjugate to te. Then N_{H}⟨t, e⟩ = C_{H}(e). So kCl(e)b = T, and by the same argument, kCl(te)b = T. If e is Gconjugate to te, then Lemma 2.4 implies that kCl(e)B = S ⊕ S. Otherwise, Lemma 2.4 implies that kCl(e)B = S and kCl(de)B = S.

Dihedral or semi-dihedral extended defect group

We prove parts (iii) and (iv) of Theorem 1.6. So assume that E is dihedral or semi-dihedral. As E splits over D, all real irreducible characters in B have Frobenius-Schur indicator +1. Let E = D⟨e⟩, where e^{2} = 1. Then E\D is a union of two conjugacy classes of E:

{d^{2m}e | m = 0, 1, …} and {d^{2m+1}e | m = 0, 1, …}.

Now C_{D}(xe) = ⟨t⟩, for each x ∈ D. As ⟨t⟩ is a minimal nontrivial subgroup of D, Corollary 18 of [14] and Lemma 2.9 imply that each Bcomponent of kΩ has vertex ⟨t⟩, and hence that each such component is isomorphic to S_{2^{n−1}}. Similarly, each b-component of kΩ_{H} is isomorphic to T_{2^{n−1}}. It then follows from Lemma 2.4 that T_{2^{n−1}} is the Green correspondent of S_{2^{n−1}} with respect to (G, D, H).

We use induction on |E| to prove that S_{2^{n−1}} occurs twice as a component of kCl(e), if e and de are conjugate in G. Otherwise it occurs once as a component of each of kCl(e) and kCl(de). The base case |E| = 4 is covered by the abelian extended defect group case.

We can apply our inductive hypothesis to \overline{b}, as (\overline{D}, \overline{E}) is a defect pair of \overline{b}, \overline{E} is a dihedral group, and |\overline{E}| < |E|. Note that e and te are conjugate in E, since t is an even power of d. So the inverse image N_{H}⟨t, e⟩ of C\overline{H}(\overline{e}) in H contains C_{H}(e) as a subgroup of index 2.

Suppose first that e is H-conjugate to de. Then \overline{e} is \overline{H}-conjugate to \overline{de}. By our inductive hypothesis, T_{2^{n−2}} occurs twice as a component of k_{C\overline{H}(\overline{e})}↑\overline{H}. The inflation of the latter module to H is k_{N_{H}⟨t,e⟩}↑^{H}. Thus T has multiplicity 2^{n−1} = 2 × 2^{n−2} as a composition factor of k_{N_{H}⟨t,e⟩}↑^{H}. Lemma 2.3 then implies that T has multiplicity 2^{n} = 2 × 2^{n−1} as a composition factor of k_{C_{H}(e)}↑^{H} = kCl_{H}(e). It follows that kCl_{H}(e) has two components that are isomorphic to T_{2^{n−1}}. We conclude from Lemma 2.4 that S_{2^{n−1}} occurs twice as a component of kCl(e).

Suppose that e is not H-conjugate to de. Then \overline{e} is not \overline{H}-conjugate to \overline{de}. By our inductive hypothesis, T_{2^{n−2}} occurs once as a component of each of k_{C\overline{H}(\overline{e})}↑\overline{H} and k_{C\overline{H}(\overline{de})}↑\overline{H}. Repeating the argument above, T_{2^{n−1}} occurs once as a component of each of kCl_{H}(e) and kCl_{H}(de). Suppose that e is G-conjugate to de. Then Lemma 2.4 implies that S_{2^{n−1}} occurs twice as a component kCl(e). Otherwise, Lemma 2.4 implies that S_{2^{n−1}} occurs once as a component of each of kCl(e) and kCl(de).

If E is dihedral, both e and te are involutions. Then kΩ has two components that belong to B. Both are isomorphic to S_{2^{n−1}}. It then follows from Lemma 2.14 that all irreducible characters in B are real. On the other hand, if E is semi-dihedral, only e is an involution. Then kΩ has one component that belongs to B. This component is isomorphic to S_{2^{n−1}}. It then follows from Lemma 2.14 that exactly half the irreducible characters in B are real.

Modular extended defect group

We prove part (v) of Theorem 1.6. So assume that E is a modular 2group i.e. that E = D⟨e⟩ where e^{2} = 1 and d^{e} = td. Then e and te are conjugate in E and are the only involutions in E\D. As C_{D}(e) = ⟨d^{2}⟩, Theorem 1.3 implies that some B-component of kΩ has vertex ⟨d^{2}⟩.

Let s be one of the two elements of order 4 in D. Then se and tse both have order 4. So \overline{e} and \overline{se} are involutions in \overline{E} that are not conjugate in \overline{H}. As \overline{b} has abelian extended defect group \overline{E}s, our proof of part (i) of Theorem 1.6 implies that T occurs once as a component of k_{C\overline{H}(\overline{e})}↑\overline{H}.

The inverse-image of C\overline{H}(\overline{e}) in H is N_{H}⟨t, e⟩. So T occurs once as a component of k_{N_{H}⟨t,e⟩}↑^{H}. As [N_{H}⟨t, e⟩ : C_{H}(e)] = 2, Lemma 2.3 implies that T occurs twice as a composition factor of kCl_{H}(e). In view of Lemma 3.3, this accounts for all b-composition factors in kΩ_{H}. Now T_{2} is the only indecomposable b-module that has at most two composition factors and vertex ⟨d^{2}⟩. So by the first paragraph, T_{2} occurs once as a component of kCl_{H}(e). Lemma 2.4 then implies that fT_{2} occurs once as a component of kCl(e). Here fT_{2} is the Green correspondent of T_{2} with respect to (G, D, H).

The previous paragraph and Lemma 2.14 imply that b has two real irreducible characters. So by Lemma 3.2 B also has two real irreducible characters. So S occurs with multiplicity 2 as a composition factor of kΩ. In particular the B-component fT_{2} of kCl(e) has at most 2 composition factors. We deduce from this and Lemma 3.1 that fT_{2} = S_{2}. Thus S_{2} is the unique component of kΩ that belongs to B.

Generalized quaternion extended defect group

We prove part (vi) of Theorem 1.6. So assume that E is a generalized quaternion group i.e. E = D⟨e⟩ where e^{2} = t and d^{e} = d^{−1}. Since E does not split over D, Lemma 1.2 implies that kΩB = 0 and ∑_{χ∈Irr(B)} ϵ(χ)χ(1) = 0. Since all irreducible characters in B have the same degree, it follows that B has an equal number of irreducible characters with Frobenius-Schur indicator +1 and −1. The same result holds for b.

Now \overline{b} has defect pair (\overline{D}, \overline{E}) and \overline{E} is a dihedral group. We deduce from part (iii) of Theorem 1.6 that all 2^{n−1} irreducible characters in \overline{b} are real with Frobenius-Schur indicator +1. So the remaining 2^{n−1} irreducible characters in b are real with Frobenius-Schur indicator −1.

Lemma 3.2 and the previous paragraph show that all irreducible characters in B are real. We conclude from the first paragraph that half the irreducible characters in B have Frobenius-Schur indicator +1 and the other half have Frobenius-Schur indicator −1.

4. Type (I) blocks with Klein-four defect group

We will prove Theorem 1.8 in this section. So assume that B is a real 2-block of G with Klein-four defect group D that is of type (I), and E is an extended defect group of B that contains D.

Recall the notation of Section 1. From the decomposition matrix of B, all four irreducible characters in B have the same degree. Set b := β^{N_{G}(D)}. Then b is the Brauer correspondent of B. There is a unique irreducible k C_{G}(D)-module T_{0} in β. As C_{G}(D) is the inertial group of T_{0} in N_{G}(D), the induced module T := T_{0}↑^{N_{G}(D)} is the unique irreducible k N_{G}(D)-module in b.

According to [4, VII], the subsections of B are (π_{1}, b_{1}), (π_{2}, b_{2}), (π_{3}, b_{3}) and (1, B). Here π_{i} is a 2-element and b_{i} is a 2-block of C_{G}(π_{i}) such that b_{i}^{G} = B. Moreover, for each i there exists an element n_{i} ∈ G such that b_{i} = (β^{n_{i}})^{C_{G}(π_{i})}. Each b_{i} has a unique irreducible Brauer character ψ_{i}, and the generalized decomposition number of each χ_{j} with respect to (π_{i}, ψ_{i}) is ±1.

Abelian extended defect group

We prove part (i) of Theorem 1.8. So assume that E ≤ C_{G}(D). Then β is a real block, by Theorem 1.4. Thus b_{1}, b_{2} and b_{3} are real blocks. We deduce from this and the fact that the generalized decomposition numbers of B are ±1 that all four irreducible characters in B are realvalued.

Suppose that E does not split over D. Then Lemma 1.2 implies that kΩB = 0 and ∑_{i=1}^{4} ϵ(χ_{i})χ_{i}(1) = 0. It follows that two of the irreducible characters in B have Frobenius-Schur indicator +1, while the other two have Frobenius-Schur indicator −1.

Suppose then that E splits over D. We apply Lemma 2.10 to N_{G}(D) and its 2-block b. Thus the k N_{G}(D)-module

M := ∑_{i≥0} T ⊗_{k} rad^{i}(kD)/rad^{i+1}(kD)

is a direct summand of kΩ_{N_{G}(D)}b and each of its components has vertex D. The factors rad^{i}(kD)/rad^{i+1}(kD) are inflated from C_{G}(D) and T is induced from a k C_{G}(D)-module. It then follows from Lemma 2.2 that M ≅ T^{4}.

Let fT be the kG-module that is the Green correspondent of T with respect to (G, D, N_{G}(D)). Then fT^{4} is a direct summand of kΩB, by Lemma 2.4. But according to Lemma 2.14, the multiplicity of S as a composition factor of kΩ is ∑_{i=1}^{4} ϵ(χ_{i}) ≤ 4. It follows that ϵ(χ_{i}) = +1, for i = 1, 2, 3, 4, and also that fT = S and kΩB = S^{4}. Finally, the O-character of kΩB is 4χ, where χ is the unique irreducible character in B which takes a positive integer value on each element of D. This completes the proof of part (i) of Theorem 1.8.

D_{8} extended defect group

We prove part (ii) of Theorem 1.8. So assume that E ≅ D_{8}. Write E = ⟨s, t, e⟩ and D = ⟨s, t⟩, where s, t and e are involutions, and s^{e} = s and t^{e} = st. Then C_{D}(xe) = ⟨s⟩, for all x ∈ D. So by Theorem 1.3 and Lemma 2.9, every component of kΩB has vertex ⟨s⟩.

Recall that χ_{B} is the G ≀ Σ-character of the lift of B to characteristic zero. Since (te)^{2} = s, Lemma 2.13 that

(4)  ∑_{i=1}^{4} ϵ(χ_{i})χ_{i}(s) is a positive integer.

Suppose we are in Case (I)1 of Lemma 1.7. Then s, t and st are conjugate in G and the subsections of B are (s, b_{1}), (s, b_{2}), (s, b_{3}) and (1, B). Equation 7.2_{1} of [4] shows that

\begin{aligned}∑_{i=1}^{4} ϵ(χ_{i})χ_{i}(s) = & & ψ_{1}(s)(ϵ(χ_{1}) + ϵ(χ_{2}) − ϵ(χ_{3}) − ϵ(χ_{4})) \\ & + & ψ_{2}(s)(ϵ(χ_{1}) − ϵ(χ_{2}) + ϵ(χ_{3}) − ϵ(χ_{4})) \\ & + & ψ_{3}(s)(ϵ(χ_{1}) − ϵ(χ_{2}) − ϵ(χ_{3}) + ϵ(χ_{4})). \end{aligned}

It then follows from (4) that at least two of χ_{1}, χ_{2}, χ_{3}, χ_{4} are not realvalued. But every real 2-block has at least one real-valued irreducible character. So we may assume that χ_{1} and χ_{2} are real, while χ_{3} = \overline{χ_{4}}.

Suppose we are in Case (I)2 of Lemma 1.7. Then s and t are not conjugate in G and the subsections of B are (s, b_{1}), (t, b_{2}), (t, b_{3}) and (1, B). Equation 7.2_{1} of [4] shows that

∑_{i=1}^{4} ϵ(χ_{i})χ_{i}(s) = ψ_{1}(s)(ϵ(χ_{1}) + ϵ(χ_{2}) − ϵ(χ_{3}) − ϵ(χ_{4}))

It then follows from (4) that at least two of the irreducible characters in B are not real-valued. So just as before we may assume that χ_{1} and χ_{2} are real-valued, while χ_{3} = \overline{χ_{4}}.

In either of the above two cases B has two real-valued irreducible characters and the decomposition matrix of B is a 4 × 1-column of 1’s. It then follows from Lemma 2.14 that S occurs twice as a composition factor of kΩ. Now S is not a component of kΩ, as it has vertex D. Thus kΩB is an indecomposable uniserial module U(S, S) whose vertex is ⟨s⟩. The O-character of kΩB can be taken to be χ_{1} + χ_{2}.

Similar arguments show that exactly one of b_{1}, b_{2}, b_{3} is a real block. Case (I)3 of Lemma 1.7 does not occur, as t and st are conjugate in

E. This completes the proof of part (ii) of Theorem 1.8.

5. Type (II) and (III) blocks with Klein-four defect group

We will prove Theorems 1.9 and 1.10 in this section. So assume that B is a real 2-block of G with Klein-four defect group D that is of type (II) or (III), and E is an extended defect group of B that contains D. Recall the notation of Section 1. Lemma 1.7 shows that B has three irreducible modules S, X and Y. In both types we may assume that S is a self-dual module. It is then clear from the decomposition matrices given in Lemma 1.7 that χ_{1} and χ_{2} are real-valued irreducible characters. Then ϵ(χ_{1}) = ϵ(χ_{2}) = +1, and ϵ(χ_{i}) ≥ 0, for i = 3, 4, by Lemma 1.5.

Set b = β^{N_{G}(D)}. Then b is the Brauer correspondent of B in N_{G}(D). Now β has a unique irreducible module T_{0}. The inertial group I of β in N_{G}(D) equals that of T_{0}, and I/ C_{G}(D) is cyclic of order 3. It follows that T_{0} has three extensions to I, which we label as S_{I}, X_{I} and Y_{I}. Then S_{1} := S_{I}↑^{N_{G}(D)}, X_{1} := X_{I}↑^{N_{G}(D)} and Y_{1} := Y_{I}↑^{N_{G}(D)} are the three irreducible k N_{G}(D)-modules that belong to b. In particular b is of type (II) or (III). We may assume that S_{I} and S_{1} are self-dual.

Abelian extended defect group

We prove part (i) of Theorems 1.9 and 1.10. Assume that E ≤ C_{G}(D). We claim that in this case E splits over D. Theorem 1.4 implies that β is a real 2-block of C_{G}(D). So T_{0} is a self-dual k C_{G}(D)- module. We may choose notation so that S_{I} is a self-dual kI-module. Then S_{I} ⊗ U = X_{I} and S_{I} ⊗ U^{∗} = Y_{I}, where U is a non-trivial 1dimensional kI/ C_{G}(D)-module. In particular Y_{1} = X_{1}^{∗}. So S_{1} is the only self-dual irreducible b-module.

Now b is a real block of type (II) or (III), with defect pair (D, E). Let χ̂_{1}, χ̂_{2}, χ̂_{3}, χ̂_{4} be the irreducible characters in b, with the notation chosen so that S_{1}, X_{1} and Y_{1} appear in the modular decomposition of χ̂_{2}, while X appears in the modular decomposition of χ̂_{3} and Y appears in the modular decomposition of χ̂_{4}. It is clear that χ̂_{1} and χ̂_{2} are real-valued. As Y_{1} is the dual of X_{1}, the characters χ̂_{3} and χ̂_{4} form a complex conjugate pair.

Let Φ_{X} be the principal indecomposable character corresponding to X_{1}. Then

ϵ(Φ_{X}) = ϵ(χ̂_{2}) + ϵ(χ̂_{3}) = ϵ(χ̂_{2}) ≠ 0.

So ϵ(χ̂_{2}) = +1. Similarly ϵ(Φ_{Y}) = 1 and ϵ(Φ_{S}) = 2 and ϵ(χ̂_{1}) = +1. It then follows from Lemma 2.14 that S_{1} occurs twice and each of X_{1} and Y_{1} occurs once, as a composition factor of kΩ_{N_{G}(D)}. We then deduce from Lemma 1.2 that E splits over D, which proves our claim.

Regard kD as a k N_{G}(D)/ C_{G}(D)-module, via the conjugation action of N_{G}(D) on D. Then

(5) kD/ rad(kD) and rad(kD)/rad^{2}(kD) are trivial N_{G}(D)-modules.

Now by direct calculation, rad(kD)/rad^{2}(kD) ≅ U ⊕ U^{∗}, as I/ C_{G}(D)- modules. Moreover, if N_{G}(D)/ C_{G}(D) ≅ Σ_{3}, then rad(kD)/rad^{2}(kD) ≅ U↑^{N_{G}(D)}, as k N_{G}(D)/ C_{G}(D)-modules. We note also that

\begin{aligned}S_{I}↑^{N_{G}(D)} ⊗ U↑^{N_{G}(D)} & ≅ & (S_{I} ⊗ U) ↑^{N_{G}(D)} & ⊕ & (S_{I} ⊗ U^{∗}) ↑^{N_{G}(D)} \\ & ≅ & X_{I}↑^{N_{G}(D)} & ⊕ & Y_{I}↑^{N_{G}(D)}. \end{aligned}

So whether N_{G}(D)/ C_{G}(D) ≅ ℤ_{3} or Σ_{3}, we have

(6)  S_{1} ⊗ rad(kD)/rad^{2}(kD) ≅ X_{1} ⊕ Y_{1}.

In the hypothesis of Lemma 2.10, take the group to be N_{G}(D) and the block to be b. As S_{1} is the only self-dual irreducible b-module, we conclude from (5), (6) and Lemma 2.10 that

S_{1} ⊕ S_{1} ⊕ X_{1} ⊕ Y_{1}

is a direct summand of kΩ_{N_{G}(D)}b.

Let f be the Green correspondence with respect to (G, D, N_{G}(D)). Lemma 2.4 and the previous paragraph imply that

(7) fS_{1} ⊕ fS_{1} ⊕ fX_{1} ⊕ fY_{1} is a direct summand of kΩB.

Now s is not a square in E, as E is an elementary abelian 2-group. It then follows from Lemma 2.13 that

(8)  ∑_{i=1}^{4} ϵ(χ_{i})χ_{i}(s) = 0.

Suppose that B is of type (II). Then (8) and Lemma 1.7 imply that

δψ_{1}(1) (ϵ(χ_{1}) − ϵ(χ_{2}) + ϵ(χ_{3}) + ϵ(χ_{4})) = 0.

Thus \overline{χ}_{3} = χ_{4}. Considering the decomposition matrix of B, Lemma 2.14 implies that S occurs twice, and both X and Y occur once, as composition factors of kΩ. It then follows from (7) that fS_{1} = S, and fX_{1} and fY_{1} are distinct irreducible B-modules, and

kΩB = S ⊕ S ⊕ X ⊕ Y.

The character of the lift of kΩB to characteristic zero is 2χ_{1} + χ_{3} + χ_{4}. Notice that χ_{3} and χ_{4} are non-real constituents of the involution module.

Suppose that B is of type (III). Then (8) and Lemma 1.7 imply that

δψ_{1}(1) (ϵ(χ_{1}) + ϵ(χ_{2}) − ϵ(χ_{3}) − ϵ(χ_{4})) = 0.

Thus ϵ(χ_{3}) = ϵ(χ_{4}) = +1, and hence X and Y are self-dual modules. Considering the decomposition matrix of B, Lemma 2.14 implies that S occurs four times, and both X and Y occur twice, as composition factors of kΩ.

From the structure of the projective indecomposable B-modules, each indecomposable B-module that is not irreducible contains S as a composition factor. Now fX_{1}^{∗} = fY_{1}, as X_{1}^{∗} = Y_{1}. In particular neither fX_{1} nor fY_{1} is irreducible. So both fX_{1} and fY_{1} contain S as a composition factor. It follows from this that S occurs at most two times as a composition factor of fS_{1} ⊕ fS_{1}.

Now X is not a direct summand of a permutation module, as it is not liftable to an RG-module. In particular fS_{1} ≅ X̸. We claim that X is not a submodule of fS_{1}. Suppose otherwise. Then X is also a quotient module of fS_{1}, as fS_{1} and X are self-dual. It follows that X occurs at least twice as a composition factor of fS_{1}. This is impossible, as X occurs at most two times as a composition factor of fS_{1} ⊕ fS_{1}. Our claim follows. In the same way, Y is not a submodule of fS_{1}.

The previous paragraph implies that S is a submodule of fS_{1}. We claim that fS_{1} = S. Suppose otherwise. Then by duality S occurs at least twice as a composition factor of fS_{1}. This is impossible, as S occurs at most two times as a composition factor of fS_{1} ⊕ fS_{1}. Our claim follows.

We claim that S is not a submodule of fX_{1}. For otherwise there is a non-projective map in Hom(fS_{1}, fX_{1}). It follows from the Green correspondence theorem that there is a nonzero homomorphism S_{1} → X_{1}. But this is nonsense, as S_{1} and X_{1} are non-isomorphic irreducible N_{G}(D)-modules. This proves our claim. By a similar argument, S is not a factor module of fX_{1}.

The previous paragraph means that we may choose notation so that X is a submodule of fY_{1}. Then by duality, X is a factor module of fX_{1}. This accounts for all occurrences of X as a composition factor of kΩ. By exhaustion, Y is a submodule of fX_{1} and a factor module of fY_{1}. This accounts for all occurrences of Y as a composition factor of kΩ. We now know that X = hd(fX_{1}) = soc(fY_{1}) and Y = hd(fY_{1}) = soc(fX_{1}). We deduce from this that fX_{1} = U(X, S, Y) and fY_{1} = U(Y, S, X).

We have shown that

kΩB = S ⊕ S ⊕ U(X, S, Y) ⊕ U(Y, S, X).

The O-character of S is χ_{1}. The O-character of U(X, S, Y) takes a positive integer value at s. So it must equal χ_{2}. Similarly U(Y, S, X) has O-character χ_{2}.

D_{8} extended defect group

We prove part (ii) of Theorems 1.9 and 1.10. So suppose that E ≅ D_{8}. Then E = D : ⟨e⟩, where e has order 2, s^{e} = s and t^{e} = st. Clearly C_{D}(xe) = ⟨s⟩, for all x ∈ D. Theorem 1.3 then implies that every component of kΩB has vertex ⟨s⟩. Moreover, as s = (te)^{2}, Lemma 2.13 implies that

(9)  ∑_{i=1}^{4} ϵ(χ_{i})χ_{i}(s) is a positive integer.

According to [4], there is a unique block b_{1} of C_{G}(s) such that b_{1}^{G} = B. Moreover, b_{1} has defect group D and extended defect group E. As s and t are not conjugate in C_{G}(s), the block b_{1} is of type (I). Let T be the unique irreducible b_{1}-module and let ψ_{1} be the Brauer character of T. Then kΩ_{C_{G}(s)}b_{1} = U(T, T), by Theorem 1.8. Lemma 2.4 now implies that kΩB is the Green correspondent of U(T, T). In particular kΩB is indecomposable.

By [2, 6.6.3] the module U(T, T) is its own Heller translate. As Green correspondence commutes with Heller translation, it follows that kΩB is its own Heller translate. So we have a short exact sequence

(10)  0 → kΩB → P → kΩB → 0,

where P is the projective cover of kΩB.

Suppose that B is of type (II). Then (9) and Lemma 1.7 imply that

δψ_{1}(1)(ϵ(χ_{1}) − ϵ(χ_{2}) + ϵ(χ_{3}) + ϵ(χ_{4})) is a positive integer.

Thus χ_{3} and χ_{4} are real-valued. From Lemma 2.14, and the decomposition matrix of B given in Lemma 1.7, we see that kΩB has two composition factors isomorphic to each of S, X and Y. Considering (10), this shows that P has four composition factors isomorphic to each of S, X and Y. It follows from this that P is the direct sum of the projective covers of S, X and Y, and hence hd(kΩB) ≅ rad(kΩB) ≅ S ⊕ X ⊕ Y.

Now End_{kG}(kΩB) is 4-dimensional; the identity map, together with the projections onto each of the submodules S, X and Y, constitute a basis. Let χ = ∑_{i=1}^{4} n_{i}χ_{i} be the O-character of kΩB. Then χ(s) = ψ_{1}(n_{1}−n_{2}+n_{3}+n_{4}) is a positive integer, while n_{1}+n_{2} = n_{2}+n_{3} = n_{2}+ n_{4} = 2. So either n_{1} = n_{2} = n_{3} = n_{4} = 1 or n_{1} = n_{3} = n_{4} = 2, n_{2} = 0. The latter case would imply that End_{kG}(kΩB) is 12-dimensional. We conclude that χ = χ_{1} + χ_{2} + χ_{3} + χ_{4}. This completes the proof of part (ii) of Theorem 1.9.

Suppose that B is of type (III). Then (9) and Lemma 1.7 imply that

δψ_{1}(1)(ϵ(χ_{1}) + ϵ(χ_{2}) − ϵ(χ_{3}) − ϵ(χ_{4})) is a positive integer.

Thus \overline{χ_{3}} = χ_{4} and δ = +1. From Lemma 2.14, and the decomposition matrix of B given in Lemma 1.7, we see that kΩB has two composition factors isomorphic to S, and one to each of X and Y. Considering (10), this shows that P has four composition factors isomorphic to S and two to each of X and Y. It follows from this that P is the projective cover of S. Thus soc(kΩB) ≅ hd(kΩB) ≅ S, and rad(kΩB)/ soc(kΩB) ≅ X ⊕Y. The character of the lift of kΩB to characteristic zero is χ_{1}+χ_{2}. This completes the proof of part (ii) of Theorem 1.10.

6. Acknowledgement

I had interesting discussions about the involution module with Wolfgang Willems, Burkhard Külshammer and Shigeo Koshitani. Their visits to NUI-Maynooth, in the Spring of 2006, were partially funded by grant RFP/2005/Mat0019, from Science Foundation Ireland. An email exchange with E. C. Dade inspired Lemma 2.10.

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Mathematics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland.

E-mail address: John.Murray@nuim.ie